= Spatial optical coherence
{title2=$g^{(1)}_{12}(0)$}
= Spatial coherence
{synonym}
<Spatial optical coherence> compares field samples at different positions, usually across a transverse plane at equal time. Two small openings separated by $d$ sample this correlation: recombine their <light> with nearly matched travel times and measure <interferometric visibility> near the central <interference fringes>. A narrow opening in front of a broad-spectrum lamp can improve transverse <spatial optical coherence> while leaving its short <temporal optical coherence>. These are separate controls: narrow spectral selection by itself need not make an extended source spatially coherent. MIT's lecture notes illustrate the two-opening test and these contrasting source preparations. https://ocw.mit.edu/courses/2-71-optics-spring-2009/bebe78c6a2ae2603a0d1a3ea85e0b4ca_MIT2_71S09_lec22.pdf[MIT 2.71/2.710, lecture 22, slides 3–5: spatial sampling and source examples]{ref}.
To isolate source width, stipulate a distant, narrowband source with uniform angular brightness on $-\Theta/2\leq\alpha\leq\Theta/2$, and mutually uncorrelated emission from distinct source directions. Here $\Theta\ll1$ is the full angular width in <radians>, $\lambda$ the common <wavelength>, and the two openings are separated by $d$ along the same transverse direction as the source extent. Neglect amplitude differences across the openings. Each source direction contributes a <phase difference> approximately $2\pi d\alpha/\lambda$. Averaging its complex cross term gives
$$
g^{(1)}_{12}(0)=\frac1\Theta\int_{-\Theta/2}^{\Theta/2}e^{i2\pi d\alpha/\lambda}\,d\alpha
=\frac{\sin(\pi d\Theta/\lambda)}{\pi d\Theta/\lambda},
\qquad \mathcal V=\left|\frac{\sin(\pi q)}{\pi q}\right|,
\qquad q=\frac{d\Theta}{\lambda}.
$$
The <sinc function> is extended continuously to one at $q=0$. The sign convention in the exponential depends on which opening is numbered first; the symmetric model gives the same real result either way. Equal detected single-opening <light intensities> make the magnitude equal to <interferometric visibility>. A displaced source center adds a deterministic <wave phase>, shifting the <interference fringes> without changing their contrast.
\Image[/optical-coherence-source-width.png]
{title=Calculated <spatial optical coherence> for a stipulated uniform incoherent angular source: wider sources wash out and invert <interference fringes>}
The strips show $I/(I_1+I_2)=1+g^{(1)}_{12}(0)\cos\phi$ with equal single-opening <light intensities>, a common <phase difference> coordinate $\phi$ and no <single-slit diffraction> envelope. At $q=1/2$, contrast is $2/\pi\approx0.636620$; at $q=1$, it vanishes. At $q=3/2$, the correlation is $-2/(3\pi)\approx-0.212207$: weak fringes return with bright and dark positions exchanged. For $\lambda=500\,\mathrm{nm}$ and $d=0.5\,\mathrm{mm}$, the first zero occurs at $\Theta=1\,\mathrm{mrad}$. This is a calculated example, not an experimental measurement or a universal source-width law.
A proposed source-width control follows directly from this model: illuminate two fixed openings with a distant adjustable luminous slit of full width $w$ at distance $R$, so $\Theta\approx w/R$ under the <small-angle approximation>, and hold $d$ and the spectral selection fixed. Increasing $w$ overlays more mutually incoherent, displaced <interference fringe> patterns. Record profiles over several near-central fringes, measure each opening separately and correct intensity imbalance as in <first-order optical coherence>. Restrict the recorded region so the path difference stays small compared with the <optical coherence length>, and resolve the fringes with the detector. Real source nonuniformity, finite opening widths and spectral spread modify the stipulated <sinc function> law. This protocol is a model-based extension of the verified two-opening method, not a claim that the plotted curves were measured.
Thus an extended incoherent source can give locally excellent <wave interference> when the sampled separation is small, and almost none when it is larger. The useful scale $d\Theta\sim\lambda$ expresses how source extent limits the aperture over which coherent addition is available. The same angular averaging explains why narrowing a source opening can recover visible <interference fringes>, even without narrowing its <optical spectrum>.
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